Exploratory Data Analysis

1. Time Series Plot

U.S. Dollar Index

library(ggplot2)
library(tidyverse)
library(plotly)

dxy <- read.csv("./data/USD.csv",header =TRUE)
dxy$Date <- as.Date(dxy$Date, format = "%m/%d/%Y")
dxy <- dxy[order(dxy$Date),]
gg <- ggplot(data = dxy, aes(x = Date, y = Price)) +   
  geom_line(color='purple') +  
  labs(x = "Year", title = "Trend of U.S. Dollar Index")

plotly_gg <- ggplotly(gg)
plotly_gg

The time series plot of the U.S. Dollar Index shows an upward trend and quarterly seasonality. However, it does not exhibit clearly periodic fluctuations across years. I preliminarily consider it to follow an multiplicative model.

Trade Balance

bea <- read.csv("./data/bea.csv")
bea$time <- as.Date(bea$time)

gg <- ggplot(bea) + 
  geom_line(aes(x = time, y = balance/1000), color = "purple") +
  labs(x = "Year", y = "Billions of Dollars", 
  title = "Trends of Trade Balance")

plotly_gg <- ggplotly(gg)
plotly_gg

The time series plot of the trade balance shows a strong seasonality, with July being the lowest month each year and January being the highest. This pattern may be due to factors such as seasonal fluctuations in demand and supply, where imports tend to increase during the holiday season in December, leading to a higher trade deficit in January. On the other hand, July may see a dip in trade activity due to summer holidays, reduced production, and lower consumer demand.

GDP

gdp <- read.csv("./data/gdp.csv")
gdp$time <- as.Date(gdp$time)
gdp$total <- gdp$consumption + gdp$investment + gdp$net_export + gdp$government
gg <- ggplot(data = gdp, aes(x = time, y = total)) +   
  geom_line(color='purple') +  
  labs(x = "Year", y = "Billions of Dollars", title = "GDP Over Time")

plotly_gg <- ggplotly(gg)
plotly_gg

The time series plot of GDP shows a steady increasing trend. Since the raw data we extracted from the BEA is seasonally adjusted at annual rates, there is no noticeable seasonality in the data. Given this, I preliminarily consider the time series to follow an additive model, where the trend is consistent over time and the variations are not influenced by seasonal patterns. This additive model helps capture the general growth trajectory without the need for seasonal adjustments.

Unemployment Rate

data_unem <- read.csv("./data/unem.csv", header=TRUE)
data_unem$time <- as.Date(data_unem$time)
gg <- ggplot(data = data_unem, aes(x = time, y = unem)) +   
  geom_line(color='purple') +  
  labs(x = "Year", y = "Rate", title = "U.S. Unemployment Rate Over Time")

plotly_gg <- ggplotly(gg)
plotly_gg

The unemployment rate increased significantly, showing both the upwarding trend and seasonality. It may need to take log-transformation in the further analysis.

CPI

data_cpi <- read.csv("./data/cpi.csv", header=TRUE)
data_cpi$time <- as.Date(data_cpi$time)
gg <- ggplot(data = data_cpi, aes(x = time, y = cpi)) +   
  geom_line(color='purple') +  
  labs(x = "Year", y = "Index", title = "U.S. CPI Over Time")

plotly_gg <- ggplotly(gg)
plotly_gg

The Consumer Price Index (CPI) also shows an upward trend and seasonality. I preliminarily consider it to follow an additive model.

S&P 500

library(quantmod)
getSymbols("^GSPC", src = "yahoo", from = "2005-01-01", to = "2024-12-31") 

[1] "GSPC"

data <- data.frame(Date = index(GSPC), 
                       Open = GSPC[, "GSPC.Open"], 
                       High = GSPC[, "GSPC.High"], 
                       Low = GSPC[, "GSPC.Low"], 
                       Close = GSPC[, "GSPC.Close"])
colnames(data) <- c("Date", "Open", "High", "Low", "Close")

figc <- data %>%
  plot_ly(x = ~Date, type = "candlestick",
          open = ~Open, close = ~Close,
          high = ~High, low = ~Low)

figc <- figc %>% layout(title = "S&P 500 Index Candlestick Plot",
                        xaxis = list(type = "date", title = "Date"),
                        yaxis = list(title = "Index Price"))
figc

The S&P 500 index shows a overall upward trend. However, it does not exhibit clearly seasonality or periodic fluctuations across years. I preliminarily consider it to follow an additive model.

Gold Price

xau <- read.csv("./data/xau.csv")
xau$Date <- as.Date(xau$Date)
ggplot(data = xau, aes(x = Date, y = Price)) +   
  geom_line(color='purple') +  
  labs(x = "Year", y = "Dollar ($)", title = "Spot Gold in US Dollar")

The plot shows an upward trend in spot gold prices, with annual seasonality. I preliminarily consider it to follow a multiplicative model.

Global Commodity Price

library(readxl)
gsci <- read.csv("./data/gsci.csv")
gsci$Date <- as.Date(gsci$Date)
ggplot(data = gsci, aes(x = Date, y = Price)) +   
  geom_line(color='purple') +  
  labs(x = "Year", y = "Dollar ($)", title = "S&P GSCI Index (USD) Over Time")

The plot shows a stable trend from 2015 to 2020, an increase from 2020 to 2022, and a decrease from 2022 to 2025. There is an annual seasonality, but no obvious cycle. I preliminarily consider it to follow a multiplicative model.

House Price

house <- read.csv("./data/house.csv", header=TRUE)
house$time <- as.Date(house$time)

gg <- ggplot(data = house, aes(x = time, y = index)) +   
  geom_line(color='purple') +  
  labs(x = "Year", y = "Index", title = "House Price Index Over Time")

plotly_gg <- ggplotly(gg)
plotly_gg

The House Price Index shows an overall upward trend. However, it does not exhibit clearly periodic fluctuations across years. I preliminarily consider it to follow an multiplicative model.

International Visitors

visitors <- read.csv("./data/visitors.csv", header=TRUE)
visitors$time <- as.Date(visitors$time)

gg <- ggplot(data = visitors, aes(x = time, y = count)) +   
  geom_line(color='purple') +  
  labs(x = "Year", y = "Number of Visitors", title = "Non-U.S. Resident Visitor Arrivals to the U.S.")

plotly_gg <- ggplotly(gg)
plotly_gg

The data on visitor arrivals to the United States shows a clear seasonal pattern, with a stable, gradual increase from 2005 to 2020. I preliminarily consider it to follow an additive model.

2. Lag plot

Based on the time series plots above, we apply log transformation to the following variables: USD Index, Unemployment Rate, Gold Price, Global Commodity Price, and Number of International Visitors. This transformation is used to stabilize variance, reduce skewness, and better model the growth patterns in these variables. Taking the logarithm helps to linearize exponential growth trends, especially for variables like Gold Price and Commodity Prices, which often exhibit high volatility and nonlinear behavior.

U.S. Dollar Index

library(forecast)

dxy_ts <- ts(log(dxy$Price), start=c(2005,1), end=c(2024,52), frequency=52)
balance_ts <- ts(bea$balance, start=c(2005,1), end=c(2024,3), frequency=4)
gdp_ts <- ts(gdp$total, start=c(2005,1), end=c(2024,3), frequency=4)
unem_ts <- ts(log(data_unem$unem), start=c(2005,1), end=c(2023,12), frequency=12)
cpi_ts <- ts(data_cpi$cpi, start=c(2005,1), end=c(2023,12), frequency=12)
sp5_ts <- ts(GSPC$GSPC.Close, start=c(2005,1), frequency=252)
xau_ts <- ts(log(xau$Price), start=c(2005,1), end=c(2024,52), frequency=52)
gsci_ts <- ts(log(gsci$Price), start=c(2014,252), frequency=252)
house_ts <- ts(house$index, start=c(2005,1), end=c(2024,4), frequency=4)
visitors_ts <- ts(log(visitors$count), start=c(2005,1), end=c(2024,12), frequency=12)

gglagplot(dxy_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("USD Index") +
          theme(axis.text.x=element_text(angle=45, hjust=1))

From the lag plot, we can see that the U.S. Dollar Index has a strong positive autocorrelation. Even at lag 9, it still shows a high autocorrelation, indicating that past values have a lasting impact on current values. This suggests a persistent trend and momentum in the time series data.

Trade Balance

gglagplot(balance_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("Balance") +
          theme(axis.text.x=element_text(angle=45, hjust=1))

From the lag plot, we can see that the Exports data has a strong positive autocorrelation in the first three lags, which then gradually weakens until it becomes very weak in lags 7-9. Moreover, since data points from different quarters are clustered together, there is no clear seasonal pattern in the serial correlation. This suggests that while past values influence current values in the short term, the effect diminishes over time.

GDP

gglagplot(gdp_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("GDP") +
          theme(axis.text.x=element_text(angle=45, hjust=1))

From the lag plot, we can see that the GDP data has a strong positive autocorrelation in the first two lags, which then gradually weakens until it becomes very weak in lags 6-9. Moreover, since data points from different quarters are clustered together, there is no clear seasonal pattern in the serial correlation. This suggests that while past values influence future values in the short term, the effect diminishes over time.

Unemployment Rate

gglagplot(unem_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("Unem Rate") +
          theme(axis.text.x=element_text(angle=45, hjust=1))

CPI

gglagplot(cpi_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("CPI") +
          theme(axis.text.x=element_text(angle=45, hjust=1))

S&P 500

gglagplot(sp5_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("Index") +
          theme(axis.text.x=element_text(angle=45, hjust=1))

Gold Price

gglagplot(xau_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("Gold Price") +
          theme(axis.text.x=element_text(angle=45, hjust=1))

From the lag plot, we can see that the Spot Gold Price has a strong positive autocorrelation. Even at lag 16, it still shows a high autocorrelation, indicating that past values have a lasting impact on current values.

Global Commodity Price

gglagplot(gsci_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("S&P GSCI Index") +
          theme(axis.text.x=element_text(angle=45, hjust=1))

From the lag plot, we can see that the S&P GSCI Index has a strong positive autocorrelation. Even at lag 16, it still shows a high autocorrelation, indicating that past values have a lasting impact on current values.

House Price

gglagplot(house_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("House Price") +
          theme(axis.text.x=element_text(angle=45, hjust=1))

International Visitors

gglagplot(visitors_ts, do.lines=FALSE) +
          xlab("Lags")+ylab("Number of Visitors") +
          theme(axis.text.x=element_text(angle=45, hjust=1))

3. Decomposition

U.S. Dollar Index

decomposed <- decompose(dxy_ts)
autoplot(decomposed, main = "Decomposition Plot for U.S. Dollar Index")

Trade Balance

decomposed <- decompose(balance_ts)
autoplot(decomposed, main = "Decomposition Plot for Trade Balance")

GDP

decomposed <- decompose(gdp_ts)
autoplot(decomposed, main = "Decomposition Plot for GDP")

Unemployment Rate

decomposed <- decompose(unem_ts)
autoplot(decomposed, main = "Decomposition Plot for Unemployment Rate")

CPI

decomposed <- decompose(cpi_ts)
autoplot(decomposed, main = "Decomposition Plot for CPI")

S&P 500

decomposed <- decompose(sp5_ts)
autoplot(decomposed, main = "Decomposition Plot for S&P 500 Index")

Gold Price

decomposed <- decompose(xau_ts, "multiplicative")
autoplot(decomposed, main = "Decomposition Plot for Gold Price")

Global Commodity Price

decomposed <- decompose(gsci_ts, "multiplicative")
autoplot(decomposed, main = "Decomposition Plot for S&P GSCI Index")

House Price

decomposed <- decompose(house_ts)
autoplot(decomposed, main = "Decomposition Plot for House Price Index")

International Visitors

decomposed <- decompose(visitors_ts)
autoplot(decomposed, main = "Decomposition Plot for Number of Visitors")

For all of the data series, these decomposition plots support the claims from Part 1 - Time Series Plot.

4. ACF and PACF Plots

U.S. Dollar Index

library(gridExtra)

acf <- ggAcf(dxy_ts)+ggtitle("ACF Plot for USD Index") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
pacf <- ggPacf(dxy_ts)+ggtitle("PACF Plot for USD Index") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(acf, pacf, nrow=2)

The ACF plot shows a very slow decay over time, meaning autocorrelation remains high for many lags and decays very slowly towards zero. The PACF plot shows a sharp drop at lag 1, and then the values will rapidly approach zero. Only the first lag is significant, and the rest are close to zero. This indicates that the U.S. Dollar Index time series is non-stationary, and it may be unit root non-stationary.

Trade Balance

acf <- ggAcf(balance_ts)+ggtitle("ACF Plot for Trade Balance") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
pacf <- ggPacf(balance_ts)+ggtitle("PACF Plot for Trade Balance") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(acf, pacf, nrow=2)

Similar to the U.S. Dollar Index time series, the ACF plot for the exports data shows a very slow decay over time, while the PACF plot drops sharply at lag 1 and then rapidly approaches zero. However, the decay in the ACF plot is slightly faster than that of a typical unit root series. This suggests that the non-stationarity in the exports time series is more likely due to a trend rather than a unit root, meaning it is trend stationary.

GDP

acf <- ggAcf(gdp_ts)+ggtitle("ACF Plot for GDP") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
pacf <- ggPacf(gdp_ts)+ggtitle("PACF Plot for GDP") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(acf, pacf, nrow=2)

The ACF plot shows a very slow decay over time, meaning autocorrelation remains high for many lags and decays very slowly towards zero. The PACF plot shows a sharp drop at lag 1, and then the values will rapidly approach zero. Only the first lag is significant, and the rest are close to zero. This indicates that the GDP time series is non-stationary, and it may be unit root non-stationary.

Unemployment Rate

acf <- ggAcf(unem_ts)+ggtitle("ACF Plot for Unemployment Rate") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
pacf <- ggPacf(unem_ts)+ggtitle("PACF Plot for Unemployment Rate") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(acf, pacf, nrow=2)

CPI

acf <- ggAcf(cpi_ts)+ggtitle("ACF Plot for CPI") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
pacf <- ggPacf(cpi_ts)+ggtitle("PACF Plot for CPI") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(acf, pacf, nrow=2)

S&P 500

acf <- ggAcf(sp5_ts)+ggtitle("ACF Plot for S&P 500 Index") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
pacf <- ggPacf(sp5_ts)+ggtitle("PACF Plot for S&P 500 Index") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(acf, pacf, nrow=2)

Gold Price

acf <- ggAcf(xau_ts)+ggtitle("ACF Plot for Gold Price") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
pacf <- ggPacf(xau_ts)+ggtitle("PACF Plot for Gold Price") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(acf, pacf, nrow=2)

The ACF plot shows a very slow decay over time, meaning autocorrelation remains high for many lags and decays very slowly towards zero. The PACF plot shows a sharp drop at lag 1, and then the values will rapidly approach zero. Only the first lag is significant, and the rest are close to zero. This indicates that the Spot Gold Price time series is non-stationary, and it may be unit root non-stationary.

Global Commodity Price

acf <- ggAcf(gsci_ts)+ggtitle("ACF Plot for S&P GSCI Index") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
pacf <- ggPacf(gsci_ts)+ggtitle("PACF Plot for S&P GSCI Index") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(acf, pacf, nrow=2)

The ACF plot shows a very slow decay over time, meaning autocorrelation remains high for many lags and decays very slowly towards zero. The PACF plot shows a sharp drop at lag 1, and then the values will rapidly approach zero. Only the first lag is significant, and the rest are close to zero. This indicates that the S&P GSCI Index is non-stationary, and it may be unit root non-stationary.

House Price

acf <- ggAcf(house_ts)+ggtitle("ACF Plot for House Price Index") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
pacf <- ggPacf(house_ts)+ggtitle("PACF Plot for House Price Index") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(acf, pacf, nrow=2)

International Visitors

acf <- ggAcf(visitors_ts)+ggtitle("ACF Plot for Number of Visitors") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
pacf <- ggPacf(visitors_ts)+ggtitle("PACF Plot for Number of Visitors") + theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(acf, pacf, nrow=2)

5. Augmented Dickey-Fuller Test

U.S. Dollar Index

library(tseries)
adf.test(dxy_ts)

    Augmented Dickey-Fuller Test

data:  dxy_ts
Dickey-Fuller = -2.6215, Lag order = 10, p-value = 0.3152
alternative hypothesis: stationary

The p-value in the ADF test for the U.S. Dollar Index is 0.31. It indicates that we fail to reject the null hypothesis at the 10% significance level. This means there is insufficient evidence to conclude that the U.S. Dollar Index is stationary. Therefore, it is likely that the U.S. Dollar Index contains a unit root and is non-stationary.

Trade Balance

adf.test(balance_ts)

    Augmented Dickey-Fuller Test

data:  balance_ts
Dickey-Fuller = -2.1425, Lag order = 4, p-value = 0.5174
alternative hypothesis: stationary

The p-value in the ADF test for the exports data is 0.12. This indicates that we fail to reject the null hypothesis at the 10% significance level, but there is still some degree of significance. Considering the conclusions from Part 4 with the ACF and PACF plots, it is possible that the ADF test is detecting non-stationarity due to a unit root. However, the exports time series may actually be trend stationary, meaning it has a trend rather than a unit root, which the ADF test cannot detect. As a result, the test fails to reject the null hypothesis, but the marginal significance suggests that the series might still exhibit some form of non-stationarity that is related to a trend rather than a unit root.

GDP

adf.test(gdp_ts)

    Augmented Dickey-Fuller Test

data:  gdp_ts
Dickey-Fuller = -0.0084786, Lag order = 4, p-value = 0.99
alternative hypothesis: stationary

The p-value in the ADF test for the GDP is 0.99. This means there is insufficient evidence to conclude that the GDP is stationary at the 10% significant level. Therefore, it is likely that the time series contains a unit root and is non-stationary.

Unemployment Rate

adf.test(unem_ts)

    Augmented Dickey-Fuller Test

data:  unem_ts
Dickey-Fuller = -2.2088, Lag order = 6, p-value = 0.4882
alternative hypothesis: stationary

CPI

adf.test(cpi_ts)

    Augmented Dickey-Fuller Test

data:  cpi_ts
Dickey-Fuller = 0.39171, Lag order = 6, p-value = 0.99
alternative hypothesis: stationary

S&P 500

adf.test(sp5_ts)

    Augmented Dickey-Fuller Test

data:  sp5_ts
Dickey-Fuller = -0.67242, Lag order = 17, p-value = 0.973
alternative hypothesis: stationary

Gold Price

adf.test(xau_ts)

    Augmented Dickey-Fuller Test

data:  xau_ts
Dickey-Fuller = -2.186, Lag order = 10, p-value = 0.4996
alternative hypothesis: stationary

The p-value in the ADF test for the Spot Gold Price is 0.93. This means there is insufficient evidence to conclude that it is stationary at the 10% significant level. Therefore, it is likely that the time series contains a unit root and is non-stationary.

Global Commodity Price

adf.test(gsci_ts)

    Augmented Dickey-Fuller Test

data:  gsci_ts
Dickey-Fuller = -2.362, Lag order = 13, p-value = 0.4251
alternative hypothesis: stationary

The p-value in the ADF test for the S&P GSCI Index is 0.48. This means there is insufficient evidence to conclude that the index is stationary at the 10% significant level. Therefore, it is likely that the time series contains a unit root and is non-stationary.

House Price

adf.test(house_ts)

    Augmented Dickey-Fuller Test

data:  house_ts
Dickey-Fuller = -1.3735, Lag order = 4, p-value = 0.832
alternative hypothesis: stationary

International Visitors

adf.test(visitors_ts)

    Augmented Dickey-Fuller Test

data:  visitors_ts
Dickey-Fuller = -2.4163, Lag order = 6, p-value = 0.4008
alternative hypothesis: stationary

The p-values in the ADF (Augmented Dickey-Fuller) tests for all variables are above 0.05, which indicates that we fail to reject the null hypothesis of the test. This suggests that the time series for all variables are non-stationary, as their statistical properties, such as mean and variance, change over time. To make the time series stationary, we need to apply differencing, which helps to remove trends and make the data more stable over time, ensuring that the statistical properties are constant. Therefore, differencing is required to achieve stationarity for further time series analysis.

6. Differencing

U.S. Dollar Index

diff1 <- ggAcf(diff(dxy_ts), 50, main="ACF of First Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
diff2 <- ggAcf(diff(dxy_ts, 2), 50, main="ACF of Second Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(diff1, diff2, nrow=2)

Trade Balance

diff1 <- ggAcf(diff(balance_ts), 50, main="ACF of First Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
diff2 <- ggAcf(diff(balance_ts, 2), 50, main="ACF of Second Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(diff1, diff2, nrow=2)

GDP

diff1 <- ggAcf(diff(gdp_ts), 50, main="ACF of First Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
diff2 <- ggAcf(diff(gdp_ts, 2), 50, main="ACF of Second Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(diff1, diff2, nrow=2)

Unemployment Rate

diff1 <- ggAcf(diff(unem_ts), 50, main="ACF of First Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
diff2 <- ggAcf(diff(unem_ts, 2), 50, main="ACF of Second Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(diff1, diff2, nrow=2)

CPI

diff1 <- ggAcf(diff(cpi_ts), 50, main="ACF of First Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
diff2 <- ggAcf(diff(cpi_ts, 2), 50, main="ACF of Second Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(diff1, diff2, nrow=2)

S&P 500

diff1 <- ggAcf(diff(sp5_ts), 50, main="ACF of First Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
diff2 <- ggAcf(diff(sp5_ts, 2), 50, main="ACF of Second Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(diff1, diff2, nrow=2)

Gold Price

diff1 <- ggAcf(diff(xau_ts), 50, main="ACF of First Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
diff2 <- ggAcf(diff(xau_ts, 2), 50, main="ACF of Second Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(diff1, diff2, nrow=2)

Global Commodity Price

diff1 <- ggAcf(diff(gsci_ts), 50, main="ACF of First Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
diff2 <- ggAcf(diff(gsci_ts, 2), 50, main="ACF of Second Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(diff1, diff2, nrow=2)

House Price

diff1 <- ggAcf(diff(house_ts), 50, main="ACF of First Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
diff2 <- ggAcf(diff(house_ts, 2), 50, main="ACF of Second Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(diff1, diff2, nrow=2)

International Visitors

diff1 <- ggAcf(diff(visitors_ts), 50, main="ACF of First Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
diff2 <- ggAcf(diff(visitors_ts, 2), 50, main="ACF of Second Differencing")+ theme_bw()+
  geom_segment(lineend = "butt", color = "#5a3196") +
    geom_hline(yintercept = 0, color = "#5a3196") 
grid.arrange(diff1, diff2, nrow=2)

7. Moving Average

U.S. Dollar Index

autoplot(dxy_ts) +
  autolayer(ma(dxy_ts, 5), series="5-MA") +
  autolayer(ma(dxy_ts, 20), series="20-MA") +
  autolayer(ma(dxy_ts, 200), series="200-MA") +
  labs(title = "Moving Average Smoothing",
       y = "Value",
       x = "Time") +
  theme_minimal()

Trade Balance

autoplot(balance_ts) +
  autolayer(ma(balance_ts, 12), series="12-MA") +
  autolayer(ma(balance_ts, 24), series="24-MA") +
  autolayer(ma(balance_ts, 36), series="36-MA") +
  labs(title = "Moving Average Smoothing",
       y = "Value",
       x = "Time") +
  theme_minimal()

GDP

autoplot(gdp_ts) +
  autolayer(ma(gdp_ts, 12), series="12-MA") +
  autolayer(ma(gdp_ts, 24), series="24-MA") +
  autolayer(ma(gdp_ts, 36), series="36-MA") +
  labs(title = "Moving Average Smoothing",
       y = "Value",
       x = "Time") +
  theme_minimal()

Unemployment Rate

autoplot(unem_ts) +
  autolayer(ma(unem_ts, 12), series="12-MA") +
  autolayer(ma(unem_ts, 24), series="24-MA") +
  autolayer(ma(unem_ts, 36), series="36-MA") +
  labs(title = "Moving Average Smoothing",
       y = "Value",
       x = "Time") +
  theme_minimal()

CPI

autoplot(cpi_ts) +
  autolayer(ma(cpi_ts, 12), series="12-MA") +
  autolayer(ma(cpi_ts, 24), series="24-MA") +
  autolayer(ma(cpi_ts, 36), series="36-MA") +
  labs(title = "Moving Average Smoothing",
       y = "Value",
       x = "Time") +
  theme_minimal()

S&P 500

autoplot(sp5_ts) +
  autolayer(ma(sp5_ts, 5), series="5-MA") +
  autolayer(ma(sp5_ts, 20), series="20-MA") +
  autolayer(ma(sp5_ts, 200), series="200-MA") +
  labs(title = "Moving Average Smoothing",
       y = "Value",
       x = "Time") +
  theme_minimal()

Gold Price

autoplot(xau_ts) +
  autolayer(ma(xau_ts, 5), series="5-MA") +
  autolayer(ma(xau_ts, 20), series="20-MA") +
  autolayer(ma(xau_ts, 200), series="200-MA") +
  labs(title = "Moving Average Smoothing",
       y = "Value",
       x = "Time") +
  theme_minimal()

Global Commodity Price

autoplot(gsci_ts) +
  autolayer(ma(gsci_ts, 5), series="5-MA") +
  autolayer(ma(gsci_ts, 20), series="20-MA") +
  autolayer(ma(gsci_ts, 200), series="200-MA") +
  labs(title = "Moving Average Smoothing",
       y = "Value",
       x = "Time") +
  theme_minimal()

House Price

autoplot(house_ts) +
  autolayer(ma(house_ts, 12), series="12-MA") +
  autolayer(ma(house_ts, 24), series="24-MA") +
  autolayer(ma(house_ts, 36), series="36-MA") +
  labs(title = "Moving Average Smoothing",
       y = "Value",
       x = "Time") +
  theme_minimal()

International Visitors

autoplot(visitors_ts) +
  autolayer(ma(visitors_ts, 12), series="12-MA") +
  autolayer(ma(visitors_ts, 24), series="24-MA") +
  autolayer(ma(visitors_ts, 36), series="36-MA") +
  labs(title = "Moving Average Smoothing",
       y = "Value",
       x = "Time") +
  theme_minimal()

:::